TY - JOUR A1 - Ly, Ibrahim T1 - A Cauchy problem for the Cauchy-Riemann operator JF - Afrika Matematika N2 - We study the Cauchy problem for a nonlinear elliptic equation with data on a piece S of the boundary surface partial derivative X. By the Cauchy problem is meant any boundary value problem for an unknown function u in a domain X with the property that the data on S, if combined with the differential equations in X, allows one to determine all derivatives of u on S by means of functional equations. In the case of real analytic data of the Cauchy problem, the existence of a local solution near S is guaranteed by the Cauchy-Kovalevskaya theorem. We discuss a variational setting of the Cauchy problem which always possesses a generalized solution. KW - nonlinear PDI KW - Cauchy problem KW - Zaremba problem Y1 - 2020 U6 - https://doi.org/10.1007/s13370-020-00810-4 SN - 1012-9405 SN - 2190-7668 VL - 32 IS - 1-2 SP - 69 EP - 76 PB - Springer CY - Heidelberg ER - TY - JOUR A1 - Bär, Christian A1 - Wafo, Roger Tagne T1 - Initial value problems for wave equations on manifolds JF - Mathematical physics, analysis and geometry : an international journal devoted to the theory and applications of analysis and geometry to physics N2 - We study the global theory of linear wave equations for sections of vector bundles over globally hyperbolic Lorentz manifolds. We introduce spaces of finite energy sections and show well-posedness of the Cauchy problem in those spaces. These spaces depend in general on the choice of a time function but it turns out that certain spaces of finite energy solutions are independent of this choice and hence invariantly defined. We also show existence and uniqueness of solutions for the Goursat problem where one prescribes initial data on a characteristic partial Cauchy hypersurface. This extends classical results due to Hormander. KW - Wave equation KW - Globally hyperbolic Lorentz manifold KW - Cauchy problem KW - Goursat problem KW - Finite energy sections Y1 - 2015 U6 - https://doi.org/10.1007/s11040-015-9176-7 SN - 1385-0172 SN - 1572-9656 VL - 18 IS - 1 PB - Springer CY - Dordrecht ER - TY - JOUR A1 - Weiss, Andrea Y. A1 - Huisinga, Wilhelm T1 - Error-controlled global sensitivity analysis of ordinary differential equations JF - Journal of computational physics N2 - We propose a novel strategy for global sensitivity analysis of ordinary differential equations. It is based on an error-controlled solution of the partial differential equation (PDE) that describes the evolution of the probability density function associated with the input uncertainty/variability. The density yields a more accurate estimate of the output uncertainty/variability, where not only some observables (such as mean and variance) but also structural properties (e.g., skewness, heavy tails, bi-modality) can be resolved up to a selected accuracy. For the adaptive solution of the PDE Cauchy problem we use the Rothe method with multiplicative error correction, which was originally developed for the solution of parabolic PDEs. We show that, unlike in parabolic problems, conservation properties necessitate a coupling of temporal and spatial accuracy to avoid accumulation of spatial approximation errors over time. We provide convergence conditions for the numerical scheme and suggest an implementation using approximate approximations for spatial discretization to efficiently resolve the coupling of temporal and spatial accuracy. The performance of the method is studied by means of low-dimensional case studies. The favorable properties of the spatial discretization technique suggest that this may be the starting point for an error-controlled sensitivity analysis in higher dimensions. KW - ODE with random initial conditions KW - Global sensitivity analysis KW - Cauchy problem KW - Error control/adaptivity KW - Rothe method KW - Approximate approximations Y1 - 2011 U6 - https://doi.org/10.1016/j.jcp.2011.05.011 SN - 0021-9991 VL - 230 IS - 17 SP - 6824 EP - 6842 PB - Elsevier CY - San Diego ER -